Carnegie Mellon University
Research Showcase @ CMU
Tepper School of Business
8-1967
Economies of Scale in Cash Balances Reconsidered
Karl Brunner
University of California - Los Angeles
Allan H. Meltzer
Carnegie Mellon University, [email protected]
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The Quarterly Journal of Economics ., 81, 3, 422-436.
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273. A Convex Approximant Method for Nonconvex Extensions of Geometric Programing,
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274. A Mathematical Model of Policy Formation in a Democratic Society, by Otto A. Davis
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276. Bias in the D1IA Caused by Stock Splits, by E. Eugene Carter and Kalman J. Cohen.
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278. Optimal Level Debt Schedules for Municipal Bonds, by Kalman J. Cohen and Frederick S.
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280. Job Vacancy Measurement, by Myron L. Joseph. The Journal of Human Resources, Fall
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281. Monetary Rules: A New Look, by M. Bronfenbrenner. The Journal of Imw and Economics,
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282. Asymmetry between Bribes and Charges: Reply, by M. I. Kamien, N. L. Schwartz, and
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283. Financial and Budgetary Problems in the Large-Scale Public EDP System, by Andrew P.
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REPRINTED FROM
T H E QUARTERLY JOURNAL OF ECONOMICS
V O L . L X X X I , AUGUST, 1 9 6 7
COPYRIGHT, 1 9 6 7 , BY THE PRESIDENT AND FELLOWS OF HARVARD COLLEGE
ECONOMIES OF SCALE IN
CASH BALANCES RECONSIDERED *
KARL BRUNNEE AND ALLAN H . MELTZER
The Baumol model, 423. —The Tobin model, 427. — Conclusion, 435.
Renewed interest in the demand for money has resulted in the
clarification of a number of issues and has focused attention more
sharply on some others. One subject of dispute has been the existence of economies of scale in the demand function for money by
business firms. A familiar version of the quantity theory of money,
M = fcF, suggests the absence of economies of scale. Our own approach, the wealth adjustment model, reaches a very similar conclusion. It implies that — to a first approximation — the demand
for money estimated from cross-sections of business firms is linear
in the logarithms and unit elastic with respect to sales or transactions. This proposition is supported by a substantial body of
evidence from cross-section and time series regressions.1
The wealth adjustment model has not yet been derived from a
more general theory of wealth or utility maximization or from a
production function. This is of particular importance since Baumol
has claimed — contrary to the quantity theory — that rational behavior implies that the demand for money by business firms will increase less than proportionally to the increase in the volume of transactions.2 The asserted conflict between the Baumol analysis and the
quantity theory or the wealth adjustment model raises a number of
questions: Are business firms irrational in the management of their
cash balances? Do the quantity theory of money and the wealth
adjustment model require irrational behavior? Or, has Baumol
drawn an incorrect or overstated inference from his theory?
A number of economists have attempted to find evidence for
* We gratefully acknowledge research support from the National Science
Foundation, the University of California, Los Angeles, Carnegie Institute of
Technology, and the University of Chicago.
1. Allan H. Meltzer, "The Demand for Money: A Cross-Section Study
of Business Firms," this Journal, LXXVII (Aug. 1963); "The Demand for
Money; The Evidence from the Time Series," Journal of Political Economy,
LXXI (June 1963). Karl Brunner and Allan H. Meltzer, "Some Further Investigations of Demand and Supply Functions for Money," Journal of Finance,
X I X (May 1964).
2. W. J. Baumol, "The Transactions Demand for Cash: An Inventory
Theoretic Approach," this Journal, LXVI (Nov. 1952). Note especially p. 550
where he argues that either the quantity theory is not based on rational behavior or it "cannot have general validity."
ECONOMIES OF SCALE IN CASH BALANCES
423
BaumoFs conclusion or for what is said to be a similar conclusion
by Tobin 8 or have disputed the evidence from the cross-section
studies.4 While we find none of these studies sufficiently convincing
to cast serious doubt on the empirical results we have obtained from
tests of the wealth adjustment model, it is useful to show that those
who seek to support BaumoFs conclusion — that cash balances are
subject to sizable economies of scale — should not expect to do so
on the basis of his theory. Indeed, we will reach the conclusion that
BaumoFs model does not imply important economies of scale in the
demand for cash balances when it is completed along the lines that
he sketches. A somewhat similar conclusion applies to the Tobin
hypothesis. Neither the Baumol nor the Tobin formulations are
inconsistent with the use of the quantity theory of money (or the
wealth adjustment model) as a first approximation to the crosssection demand for money by business firms.
T H E BAUMOL MODEL
Baumol derives the transaction demand for money equation by
minimizing the cost of holding cash balances at given interest rates
and transaction costs. It is clear from his presentation, though not
always recognized, that three separate cases are considered. In the
first two, firms cannot receive cash by selling output. Cash is obtained only by borrowing or by selling financial assets. In these
two restricted cases, the familiar square root formula for optimal
cash balances is obtained.5 But it should be noted that difficulties
3. James Tobin, "The Interest Elasticity of the Transactions Demand
for Money," Review of Economics and Statistics, X X X V I I I (Aug. 1956).
Tobin is concerned primarily with the interest elasticity of transaction
balances and not with the question of economies of scale. However, Tobin
states (p. 247) that his equation will produce "essentially the same results as
Baumol s square root formula or the more complete Baumol hypothesis considered here. This statement misled one author of this paper (Meltzer "A
Cross-Section Study," op. tit.) and has apparently misled others into' the
belief that the Tobin equation suggested the importance of scale economies
m cash balances.
4. Alan W. Heston, "An Empirical Study of Cash, Securities and Other
Vummt Accounts of Large Corporations," Yale Economic Essays, Vol. 2
(1962). W. J. Frazer, Jr., "The Financial Structure of Manufacturing Corporations and the Demand for Money; Some Empirical Findings," Journal of
Political Economy, LXXII (April 1964). G. S. Maddala and R. Vogel "The
Demand for Money: A Cross-Section Study of Business Firms: Comment,"
this Journal,XXXIX (Feb. 1965); a paper with the same title by E. L. Whalen
ibid. ft. L. Teigen, /Demand and Supply Functions for Money in the United
States, Eoonometnca, Vol. 32 (Oct. 1964). Moreover, frequent references to
this conclusion can be found in the literature and have recently started to
appear m textbooks, e.g., M. L. Burstein, Money (Cambridge, Mass.: Schenkman, 1963), pp. 451^58.
5. For a discussion of some of the problems of applying the lot-size
424
QUARTERLY JOURNAL OF ECONOMICS
occur when an attempt is made to specify the time period to which
the analysis pertains or to aggregate over several firms. The simple
model is an incomplete statement of individual firm behavior and is
unsuitable for aggregation. The problem is that firms are not permitted to receive money from sales or payments from accounts receivable, a statement that is unlikely to be true if the analysis is
applied to more than a short time span or to more than a single firm.
To put the matter succinctly, it is most likely that the larger the
number of firms over which one aggregates, the smaller the time
periods to which the particular conclusion of Baumol's analysis
pertains. Even for a single firm, the time period during which there
are no receipts from sales or receivables is likely to be no more than
a few days and certainly less than the monthly or quarterly dates
at which we observe cash balances. To avoid the problem, some
statement must be made about the period to which the simple model
applies or about the money holding behavior of firms during periods
in which there are cash receipts from sales.
Baumol does not overlook the latter case, although many others
(including the second-named author of this paper) have done so
when discussing his conclusion. However, he concludes that a
similar transaction demand function is obtained if firms are permitted to receive cash at the beginning of a transaction period.
Specifically, he obtains 6 for this case
(1)
R = C + T(kw +
kd)/i
where
R = the number of dollars of receipts from sales that are not
invested in securities
C = the amount withdrawn from investment balances (dollars)
T = the number of dollars (actual or expected) used to make
payments during the period between receipts
i = the interest rate on a loan or the opportunity cost of
holding money rather than financial assets
kw = the marginal cost of withdrawing cash
kd = the marginal cost of depositing cash.
Earlier, he had obtained the familiar result that
(2)
C = y/2bwT/i (bw = the fixed cost of withdrawing cash)
formula to cash balances see W. Beranek, "Some Implied Assumptions of
Standard Inventory Models," forthcoming.
. . . .
6. Baumol, op. citp. 549. Baumol obtains this equation by minimizing
a cost function with respect to / , the number of dollars invested from the
proceeds of sales, and with respect to C, the number withdrawn (or withheld)
from investment balances.
ECONOMIES OF SCALE IN CASH BALANCES
425
applicable to the period in which there are no receipts from sales.
Since part of the money received is held in cash and since he assumes no additional receipts, equation (2) may be substituted into
the formula for R to obtain, as Baumol suggests:
(3)
R = y/SbJV% + T(kw +
kd)/i.
This procedure raises two questions of interest. First, why do
firms ever hold "speculative" balances? Second, what is the amount
of the cash balance that we should expect to observe if we apply the
Baumol analysis to a sample of firms? We consider each of these
questions in turn.
Baumol notes7 that "any receipts exceeding anticipated disbursements will be invested, since, eventually, interest earnings must
exceed ('brokerage') costs of investment." This statement is difficult to reconcile with his later observation that suggests the existence
of speculative balances. Either money balances are identical with
transaction balances or BaumoPs model is incomplete and requires
supplementary statements to describe the division of money balances
between speculative and transaction balances. Moreover, his discussion is in terms of known receipts and expected disbursements, a
probabilistic statement that seems to include the precautionary
motive as well.8 Despite BaumoPs statements to the contrary, the
analysis appears to merge a firm's "transaction balances" with
total money balances. Our evidence suggests that the merger is
appropriate, that better results are obtained if money balances are
not separated according to "motives." 9
Turning to the amount of cash that we would expect to observe,
on BaumoPs assumptions, we must distinguish two separate components of the average cash balance. One is the amount that optimizing firms will withhold from investment. During a part of the
transaction period, with length R/T = (T — I)/T, optimizing firms
will hold an average amount R/2 in cash. From equation (1),
above, it is clear that this amount includes a component C that is
described1 as the amount initially withheld from investment. The
second component is the average amount, C/2, held during the
period I/T in which there are no receipts from sales. The average
money balance for a firm must then be the weighted average of
C/2 and R/2 with weights given by the proportion of the period
7. Ibid., p. 547.
8. See also the discussion in BaumoPs fn. 2, p. 546, where various risks are
incorporated into b and i.
9. Brunner and Meltzer, "Predicting Velocity: Implications for Theory
and Policy," Journal of Finance, XVIII (May 1963).
1. Baumol, op. cit.t p. 549.
426
QUARTERLY JOURNAL OF ECONOMICS
during which we should expect to observe these balances if the
Baumol analysis is used to determine a firm's cash balance. We
denote the weighted average balance by M:
M = (R/2)[(T
- I)/T]
+ (C/2)
(I/T).
Substituting for R from equation (1), and recalling that R = T — 7,
we obtain
M = C/2 + (R/2i)(kw
+ kd).
Again substituting for R from equation (1) and rearranging
terms, this becomes
(4)
M = VOV2i
[1 + iK + kd)/i] + T/2 [ (kw +
kd)/i\*.
It should be clear that the often used square root formula is a
particular case of equation (4), the case in which kw and kd are
zero. As Baumol notes,2 this assumption becomes troublesome once
firms are permitted to receive cash from sales or repayment of accounts receivables. If the marginal cost of investing and withdrawing cash is zero, it pays to invest in non-cash assets at any positive
interest rate and for any arbitrarily short period of time. Thus the
familiar square root formula for optimal money balances cannot
be obtained under these conditions since we cannot assume kd and
kw = 0. Instead, the Baumol assumptions imply the average
(optimal) balance is a quadratic function of the volume of transactions.
However, we should note that equation (4) is not necessarily inconsistent with the quantity theory of money as an explanation of
a firm's demand for money. Two cases may be distinguished. First,
if the fixed cost of withdrawing cash, bw, is zero, the square root
term vanishes, and the desired balance becomes a linear function of
the volume of transactions with a coefficient depending on the
marginal cost of depositing and withdrawing cash and the interest
return from holding financial assets. The firm's cash balance increases linearly with the volume of transactions, interest rates and
marginal costs remaining unchanged. Moreover, for small values
of bw, the quantity theory remains a reasonable first approximation
as the evidence to which we have referred seems to suggest.
Second, for a given value of bw, the existence of economies of scale
is unlikely to be detected in the cash balances of largefirmsbehaving
according to the Baumol model of rational behavior. The elasticity
of M with respect to T, denoted e(M, T), is
(5)
*(M,T) =
[1 + (kw + kd)/i] VbwT/2i
2 [1 + (K + kd)/i] y/bwT/2i
2. Ibid., in. 6, p. 649.
+ [(kw +
kd)/i]*T/
+[(kw
kd)/i]2T.
+
ECONOMIES OF SCALE IN CASH BALANCES
427
As T approaches zero, the elasticity approaches one-half, the result
obtained from the simpler Baumol model (equation 2) for all values
of T. But for large T, the elasticity in equation (5) approaches one,
the value expected from the quantity theory. Once receipts of cash
from sales or accounts receivables are incorporated in the model,
the value of the elasticity increases toward unity with increases in
the volume of transactions. Unless bw is extremely large, the elasticity approaches unity quite rapidly as T rises, since (kw + kd) /i
is likely to be a small fraction.
However, speculation about the values of the parameters is unnecessary. Parameter estimates have been reported for rgressions
based on industry subaggregate and individual firm data.8 These
data suggest that economies of scale are quite minor and hard to
detect for firms with transactions or sales greater than $25 or $50
million per year, even if they occur up to that point. Moreover,
the data reject the often repeated implication of the simpler
Baumol hypothesis since they suggest that, where there are economies of scale in holding cash balances, the economies are exhausted
at relatively low values of transactions volume.
Thus the Baumol model provides little reason for abandoning
the earlier conclusion that to a first approximation the quantity
theory explains the observed cash holdings of business firms when
the observations are from cross sections at a point of time. It is
only when we are concerned with the behavior of relatively small
firms or the distribution of cash balances within industries that the
economies or diseconomies of scale implied by the Baumol model
may become apparent. The difference between the cash balances
expected under the Baumol model and the quantity theory is
sufficiently small that, even if the Baumol model is correct, it provides little basis for rejecting the quantity theory or concluding
that sizable economies of scale are to be expected.
T H E TOBIN MODEL
In Tobin's approach 4 the firm maximizes the net revenue from
total transaction balances assumed to consist of bonds and money.
T h e e l a s i i c i t y of cash with respect to sales was computed for fourteen
• J
industry groups in each of nine years. The cross sections are based principally
on data from Statistics of Income, U.S., Internal Revenue Service. The mean
value of the elasticity for each ranges from 1.01 to 1.16. The mean of the 126
regressions is 1.04. Eighty per cent of the computed elasticities are above
unity. For further details and tests of alternative functions, including the
quadratic, see Meltzer, "A Cross-Section Study/' op. cit.
4. Op. cit.
428
QUARTERLY JOURNAL OF ECONOMICS
Transaction balances are said to be held as a means of bridging the
gap between receipts that occur at discrete intervals and continuous
expenditures. Tobin does not specify the determinants of total
money holdings or of the distribution of money balances between
the transaction and nontransaction components. His principal concern is to show that interest rates will affect the amount of transaction balances held in the form of money.
Nevertheless, Tobin's analysis has been used to suggest that
there may be substantial economies of scale in transaction balances:
of aggregate or individual firms, In this section, we show that to a
first approximation each firm, operating according to the Tobin
model, should regard its demand function for (transactions) money
as unit elastic in the volume of transactions. We then discuss some
problems that arise if the Tobin model is applied to firms in the
aggregate. Only those portions of Tobin's analysis that are required by the present discussion will be reproduced.
Tobin solves 5 for the average amount of transactions balances
that is held in bonds rather than money. This amount is
B = (n-
l ) / 2 n [Y (1 - 4b 2 /r>)] n
2,r ^ 2b
where
B = the average amount of bonds held
n = the number of transactions made to exchange money
for bonds (restricted to integer values)
Y = the value of receipts at the beginning of the period
(equal to the amount disbursed during the period)
5 = the variable cost of exchanging money for bonds or
bonds for money
r = the interest rate paid on bonds.
He had previously defined
C = 7 / 2 — B (C = the average cash balance)
so that
(6)
C = (Y/2) [1 -
(n - 1 )/n (1 - 4b 2 /r 2 ) ] n ^ 2, r ^ 2b.
We note,first,that up to the point at which it pays to make two
transactions, all receipts (Y) are held in the form of cash; cash
balances increase in direct proportion to the volume of transactions.
C = (1/2) y. Second, as n becomes relatively large, cash balances
increase in direct proportion to the volume of transactions (or
receipts) given b and r. Third, for constant values of n, the same
conclusion is reached. But it is an essential part of the Tobin
5. Ibid., p. 245.
ECONOMIES OF SCALE IN CASH BALANCES
429
analysis that n is variable. The existence of economies of scale in
the holding of cash thus appears to result solely from the effect of
variations in n. This effect will be examined presently.
Before doing so, it is useful to investigate the expected value
of the transactions elasticity of cash balances, denoted e{C, Y),
ignoring the effect of Y on n. Differentiating equation (6) shows
that
ZC/-QY = ( 1 / 2 ) [1 -
(» -
l)/n
(1 -
4b2A2) ]
and that C/Y = ZC/dY so that e{C, Y) equals unity under the
Tobin analysis, just as it would under the quantity theory. Note
that e(C, Y) is independent of n, the number of exchanges of bonds
for money. Thus, we are led by the Tobin analysis, to expect
neither economies nor diseconomies of scale in the holding of money
balances, if changes in n are ignored.
There remains the problem of investigating the importance of
scale economies when n is permitted to vary. Tobin makes the
optimal number of transactions, n*, dependent on Y, r, b, and a
(a = the constant fixed cost component of transactions costs); n*
increases directly with Y and r and inversely with a and b. A
solution for n* can be obtained from Tobin's analysis. Optimal
n (= n*) is found when the marginal net revenue is not less than the
(constant) marginal cost of increasing the number of asset exchanges,6 i. e., when
(7)
n*(n*
+ 1) ^ Yr( 1 -
2b/r)2/2a.
However, Tobin restricts n and n* to integer values to maintain the
equality of cash receipts and expenditures for the transaction period.
It is clear, therefore, that for any given values of a, 6, and r there
will be no change in n* unless Y changes by an amount sufficient to
raise the optimal number of transactions above the preexisting n*.
Between the points at which these discrete changes take place, cash
balances are unit elastic with respect to the volume of transactions
as our previous analysis implies. Each time the optimal number of
transactions increases by one unit, the optimal cash balance shifts
down by a finite amount dependent on r and b. The relation is
again unit elastic until the next point at which the optimal number
of transactions increases. This relation is represented in Figure I.
The extent to which an individual firm's demand for money will
depart from unit elasticity depends (in the Tobin analysis) on the
frequency with which a firm observes changes in receipts or expenditures that are sufficiently large to cause the firm to cross one of the
6. Ibid., p. 245, equation (11).
430
QUARTERLY JOURNAL OF ECONOMICS
FIGURE I
"jump" points at which n* increases. Tobin is not directly concerned with the problem of economies of scale, does not solve for
the cash balance, and does not mention the jump points in optimal
cash balances that arise from the restriction on the optimal number
of transactions to an integer value per transaction period. But his
analysis can be used to compute the amount by which receipts must
change before there is a change in the optimal number of transactions between money and bonds. He shows that the number of
asset exchanges does not exceed two per period unless the requirements for his Case IV are met,7 i.e., until
(8)
(1/12) Yr (1 - 2b/r)2 è a.
From equation (7) we obtain the amount by which receipts must
change (A30 to raise or lower the optimal number of transactions.
For given a, b and r, the condition is
An* (2 n* + An* + 1) ^ A y (r/2a) (1 - 2 b/r)2
A jump point is crossed if An* = 1, i.e., if
AY ^ [4a(n* + l ) ] / r ( l - 2b/r)2.
Assume that a firm has reached the minimum level of receipts
that satisfies equation (8). For n* to increase by one unit, the firm
must experience a chapge in receipts
7. Ibid., p. 245.
431
ECONOMIES OF SCALE IN CASH BALANCES
(9)
AY = (1/3) (n* + 1) Y^
where Y^ is the minimum value of Y at which n* = 2.
When n* = 2, AY = ymln. Unless receipts double from one
transaction period to the next, a firm that is just above the threshold level of sales at which n* becomes 2 need not be concerned
with the possibility that n* may become 3. And a firm that is midway between n* = 2 and n* = 3 must experience a 60 per cent increase in sales before it crosses another jump point and experiences
additional economies of scale.
Table I shows the variation in receipts per transaction period
that is consistent with an unchanged value of n*. The values in the
table are obtained from equation (9) and are used to mark the
"jump points" in Figure I. Each time n* increases by one unit, the
distance between jump points increases by 1/3 Ymin. The minimum
value of Y that is the lower bound for each successive value of n*
increases also, so that the change in receipts required to move a
firm from one optimal n to the next becomes a smaller proportion of
the level of receipts that mark the entrance to a particular range.
Nevertheless, column (3) of the table shows that the percentage
change in receipts per period consistent with retaining a constant
value of n* remains relatively large as n* increases.
A continuation of the table would more clearly reveal the pattern, viz., as n* doubles, the values in column (3) are reduced by
50 per cent. But this information tells us only that each firm that
is not close to a jump point may treat n* as a constant up to relatively large values of n* and proceed on the assumption that cash
balances are unit elastic in the volume of transactions. The table
TABLE
I
VARIATIONS IN RECEIPTS CONSISTENT WITH AN UNCHANGED OPTIMAL
NUMBER OF TRANSACTIONS PER TRANSACTION PERIOD
Value
» * of
(1)
Minimum value of Y
at lower bound
(2)
2
Km In
3
27min
4
5
1 0 / 3 Fm In
5F„,»
6
7 Km In
etc.
etc.
Percentage of
lower bound
by which
receipts may
change without
increasing n*
(3)
100%
2/3 = 66.7%
1 / 2 = 60
%
6 / 1 5 = 40
%
1/3 = 33.3%
etc.
432
QUARTERLY
JOURNAL OF ECONOMICS
provides no basis for concluding that economies of scale are exhausted at low values of n*.
The problem is that the Tobin model gives no information from
which the length of the transaction period (or the value of Ymin)
can be obtained. But the length of the transaction period plays a
critical role in the interpretation of the analysis and particularly
so for the question of economies of scale. Without information
about the length of the transaction period, nothing can be said about
the size of scale economies. For example, if the transaction period
is a week, a firm that makes five asset exchanges probably exhausts
most of the economies of scale that may exist. Such a firm engages
in securities transactions every day. Even if receipts vary by more
than the 40 per cent per week required to move the firm from the
lower bound at which n* = 5 to n* = 6, few additional economies
of scale can be expected at the increased n*. This conclusion does
not follow if n* remains at five and the transaction period is a
month. Moreover, the conclusion does not follow if the firm makes
semiweekly transactions (n* = 26) and the transactions period is a
calendar quarter. Additional economies of scale would be expected
at values of n* larger than 26 if the transaction period is three
months. And smaller percentage changes in sales would be required
to move the firm's receipts into the range at which the additional
economies could be realized.
However, as n* increases additions to n* produce smaller downward shifts in the curve at each of the successive jump points. This
conclusion is independent of any assumption about the length of
the transaction period. A similar conclusion holds for the average
cash balance. But the absolute size of the reduction in the average
cash balance cannot be obtained from the analysis Tobin presents.
Assumptions must be introduced about the values of b and r. However, Figure I suggests the relative size of the reductions in log C
as n* changes.
Table II shows the computed values of the elasticity and the
average cash balance for successive ranges of the curve and for one
set of assumed values: b = .01, r = .04. The elasticities shown in
the table are minimum values for n* = 2 . . . 7 and arbitrarily
chosen values of b and r. The computations are based on the assumption that a firm moves along a curve connecting the cash balance held when the receipts from sales are the minimum amount at
which n* = 2 to the cash balance that is held when receipts are just
sufficient to make n* = 3, etc. The Tobin analysis denies that this
is the path that the firm follows and implies instead that a firm
ECONOMIES
OF SCALE
IN
CASH
BALANCES
433
moves between jump points along a curve that is unit elastic. If the
Tobin hypothesis is correct, values of the elasticity computed by
the method used to construct the series in the table will be underestimates of the elasticities obtained from econometric studies for
all values of 0 < b < 2r. The size of the underestimate depends on
the values of b and r. But whatever the values chosen, the minimum
values of the elasticities approach unity as n* increases. It is likely,
therefore, that even if the Tobin hypothesis is true, it may be
difficult to distinguish between this model and the quantity theory
in cross-section regressions. The data in the table suggest that the
elasticity computed by regression will be close to unity if the true
values of b and r are in the neighborhood of those we have chosen.
T A B L E II
VALUES OF THE AVERAGE CASH BALANCE AND OF THE
MINIMUM
ELASTICITY OF CASH BALANCES BETWEEN VALUES OF N*
b = .01
Y
**
2
Fmln
3
4
2 Km In
10/3 F m In
5
6
5 F m In
7 Fmln
7
2 8 / 3 Fm In
* Computed from
r = .04
c
AC
5 / 1 6 Fmln
1 / 2 Fmln
5 / 1 6 Fmln
3/16Fmln
11/ 48 Fmln
13/ 48 Fmln
35/48Fm In
Fmln
63/48FmIn
5 / 3 Fmln
15/48 Fmln
17/48 Fmln
c(C, F)*
C/Y
5/16
3/4
11/14
13/16
15/18
17/20
1/4
7/32
1/5
3/16
5/28
(AC/CXY/AY).
We may summarize our analysis up to this point by noting that
the Tobin model suggests that each firm that is not close to a jump
point may assume that its cash balance is unit elastic with respect
to the volume of transactions. We noted earlier that the same conclusion holds when the volume of transactions is relatively large or
when it is so small that n* is less than 2. To a first approximation
every firm may consider its demand function to be unit elastic. The
error in the approximation will be quite small in general and will
decline with the number of exchanges between bonds and money.
In short, there is little conflict between the implications of the
Tobin analysis and those of the quantity theory when both are
applied to individual firms.
While the conclusion just reached is valid for each firm individually, we have found that it need not hold for the aggregate of
firms or households under the Tobin analysis. Tobin provides no
clues that would help to determine the expected size of the reduction in average cash balances if large and small firms are compared.
434
QUARTERLY JOURNAL OF ECONOMICS
Of particular importance is the absence of a statement about the
length of the transaction period, the time between cash inflows.
Without this information, we do not know whether the transaction
period is assumed to be the same for firms of different size, so we
cannot compare observed average cash balances. Moreover, we do
not know the value of b, so we cannot compute the magnitude of
the economies of scale between size groups from equation (6).
However, the data from Statistics of Income can be used in one
of two ways to check the implication of the Tobin analysis and to
bring out the problem in interpretation. Equation (6) implies that
the ratio of cash to the volume of transactions per transaction period
should be 1/2 for firms with the smallest volume of transactions,
since such firms may be assumed to make no asset exchanges. (1)
Assume that the length of a transaction period is the same for all
firms. If there are economies of scale, the ratio of cash to sales
should fall with size of firm. During the years 1955-61, the ratio of
cash to sales was lower for (total) manufacturing firms of smallest
size (less than $25,000 in assets) than for any size class up to the
largest (assets in excess of $250 million). Firms in the top asset
size class have lower cash/sales ratios than those in the bottom
size class in only two of the seven years.8 These observations reject either the implications of the Tobin hypothesis with respect to
economies of scale or the assumption of uniform transaction periods.
(2) Alternatively, assume that the length of the transaction period
differs among firms in different size classes. The data for 1955-61
suggest that the average ratio of cash to annual sales for firms of
smallest size is .038. For firms of largest size, this ratio is .042. The
data can be used with equation (6) to illustrate the effect of economies of scale on the length of the transaction period. If there are
no exchanges of bonds for money or of money for bonds, the ratio
of cash to receipts (or sales) per transaction period is .50. Since
firms of smallest size may be assumed to make no asset exchanges, a
transaction period of approximately 4 weeks (.50/.038) is required
to make the data consistent with the Tobin hypothesis. If firms of
largest size make no asset exchanges per transaction period, the ratio
of cash to receipts per transaction period is .50 for them also, and
their transaction period is 1 month (.50/.042), slightly longer than
the transaction period for firms of smallest size. Inspection of
equation (6) shows that any increase in the number of asset ex8 Similar findings for individual industries may be observed in the
data published by R. Selden, "The Postwar Rise in the Velocity of Money:
A Sectoral Analysis," Journal of Finance, XVI (Dec. 1961).
ECONOMIES
OF SCALE IN CASH BALANCES
435
changes reduces the ratio of cash to receipts per transaction period
and thus increases the length of the transaction period required to
make the Tobin hypothesis consistent with the annual data.
If we are unwilling to accept the finding that the largest firms
receive cash less frequently than the smallest firms, one or two conclusions must be rejected: (1) the Tobin model implies economies
of scale; or (2) Tobin's model is applicable to the cross-section
observations. The data suggest that it is the latter conclusion that
must be rejected. The ratio of cash to receipts per transaction
period (.50) may be assumed constant (n* = 0) for firms with total
assets of less than $25,000. For the data to be consistent with the
Tobin model, the transaction period for smallfirmsmust be variable
since the ratio of cash to annual sales is variable. It is not unlikely
that the transaction period for firms in other size classes varies also.
Variations in payments schedules in turn suggest that firms search
for an optimum payments schedule and that the determination of
optimum money balances is obtained as part of a larger optimizing
problem.
The problem may be only that we have pushed the analysis too
far. Tobin was interested primarily in showing that an individual
or firm's demand for transaction balances is (beyond a minimum
value of transactions) dependent on interest rates. He stressed
neither the aggregate relationship nor the importance of economies
of scale. In any case, his analysis does not seem to be consistent
with the cross-section observations for industries unless some additional statements are introduced to explain differences in the
length of transaction periods for firms that differ in size. And his
analysis does not suggest the importance of economies of scale for
individual firms with the perhaps minor exception of those that are
close to a "jump" point.
CONCLUSION
There is probably little need to point out that the existence of
economies of scale in the demand for money is an empirical question. We have not been primarily concerned with that question
here since the evidence suggesting that scale economies are minor
or nonobservable has been presented elsewhere. The issue has been
whether or not "rational behavior" implies that there should be
economies of scale in the cross-section demand for money by firms
or individuals or in the demand for money by individual firms over
time.
436
QUARTERLY
JOURNAL OF ECONOMICS
It has now been established that rational behavior of the
Baumol or Tobin type does not deny that the quantity theory of
money explains the cross-section demand for money by firms (at
least the transaction component) to a first approximation. This, of
course, does not mean that the quantity theory provides a good explanation; both models may be false. But now the circle is complete,
since we started with the observation that, to a first approximation,
the quantity theory explained the cross-section observations quite
well.
Thus it appears that for an individual firm, the Baumol and
Tobin models are not alternatives to the familiar quantity theory.
They both imply it and as such perhaps furnish a firmer foundation
for it. The same conclusion holds under the Baumol model for
firms of different size at a particular point in time, while the Tobin
model is not sufficiently specified to permit a conclusion to be drawn
for the aggregate.
Of course, nothing that we said here should suggest that economies of scale are inconsistent with some other model based on
rational behavior that might be constructed. But analysis and
evidence seem to indicate that an extended search for substantial
economies or the construction of theories that imply them will not
prove fruitful, if the assumption of fixed payments schedules is
retained.
O H I O STATE UNIVERSITY
CARNEGIE INSTITUTE OP TECHNOLOGY
(Continued)
286. Some Network Characterization« for Mathematical Programming and Accounting
Approaches to Planning and Control, by A. Chames and W. W. Cooper. The Accounting
Review, January 1967.
287. Risk Orientation as a Predictor in the Prisoners Dilemma, by F. Trenery Dolbear, Jr.
and Lester B. Lave. The Journal of Conflict Resolution, December 1966.
288. Models and Modelling for Manpower Planning, by W. R. Dill, D. P. Gaver, and W. L.
Weber. Management Science, December 1966.
289. The Benefits and Costs of Bank Mergers, by Kaiman J. Cohen and Samuel Richardson Reid
Journal of Financial and Quantitative Analysis, December 1966.
290. Regression Yield Curves for U. S. Government Securities, by Kaiman J. Cohen, Robert L.
Kramer, and W. Howard Waugh. Management Science, December 1966.
291. Inconsistent Behavior in Lottery Choice Experiments, by F. Trenery Dolbear, Jr., and
Lester B. Lave. Behavioral Science, January 1967.
292. Strategy, Organization Planning, and Changing Problems of Corporate Management, by
Richard G. Brandenburg. Kommunikation, November 1966.
293. Analysis of Retailer Advertising Behavior, by William F. Massy and Ronald E. Frank.
Journal of Marketing Research, November 1966.
294. Private and Public Consumption and Savings in the von Neumann Model of an Expanding Economy, by Oskar Morgenstern and Gerald L. Thompson. Kyklos, 1967.
295. Optimal Investment Policy and the Flexible Accelerator, by Robert E. Lucas, Jr.
International Economic Review, February 1967.
296. Motivational and Emotional Controls of Cognition, by Herbert A. Simon. Psychological
Review, January 1967.
297. The Business School, A Problem in Organizational Design, by Herbert A. Simon. The
Journal of Management Studies, February 1967.
298. Decomposition of Communication Networks, by Kenneth D. Mackenzie. Journal of
Mathematical Psychology, February 1967.
299. The Use of Information Processing Languages in Psychology, by Herbert A. Simon.
Les Modèles et la Formalisation du Comportement. Proceedings of the Colloques Internationaux Du Centre National De La Recherche Scientifique, July 1965.
300. The Possibility of Oversupply of Local "Public" Goods: A Critical Note, by William C.
Bramara and F. Trenery Dolbear, Jr. The Journal of Political Economy, February 1967.
301. A Program of Research in Business Planning, by H. Igor Ansoff and Richard C.
Brandenburg. Management Science, February 1967.
302. On the Theory of Optimum Externality, by F. Trenery Dolbear, Jr. The American
Economic Review, March 1967.
303. A Note on Supplemental Appropriations in the Federal Budgetary Process (1) by
Gary W. Bowman, Otto A. Davis, Henry J. Gailliot, and Alan C. Hess. Papers on
Non-Market Decision Making, 1967.
304. The Job of a College President, by Herbert Simon. The Educational Record, Winter
1967.
305. An Empirical Evaluation of Alternative Portfolio-Selection Models, by Kaiman J. Cohen
and Jerry A. Pogue. The Journal of Business of the University of Chicago, April 1967.
306. "Hard" and "Soft" Lines in Economic Development, by M. Bronfenbrenner. The Libyan
Economic and Business Review, 1967.
307. Programs as Factors of Production, by Herbert A. Simon. Proceedings of the Nineteenth
Annual Winter Meeting Industrial Relations Research Association, 1967.
308. Monopoly Rents, Wage Rates, and Union Wage Effectiveness, by Leonard A. Rapping.
The Quarterly Review of Economics ir Business, Spring 1967.
309. Stock Averages, Stock Splits, and Bias, by E. Eugene Carter and Kaiman J. Cohen.
Financial Analysts Journal, May-June 1967.
310. Copper Prices; A Case Study of Guidepost Policy, by Robert C. Blattberg and Timothy
W. McGuire. California Management Review, 1967.
311. Need and Value Shifts in College Training Groups, by Howard Baumgartel and Joel W
Goldstein. The Journal of Applied Behavioral Science, 1967.
312. The Use of Simulation in Selecting Branch Banks, by E. Eugene Carter and Kaiman J.
Cohen. Industrial Management Review, Spring 1967.
313. Money Supply Revisited: A Review Article, by Allan H. Meitzer. The Journal of Political
Economy, April 1967.
314. Operations Research in the Design of Management Information Systems, by Charles H.
Kriebel. Operations Research and the Design of Management Information Systems. John
F. Pierce Jr., Editor, Fall 1966.
( Continued on back cover)
The present series begins with articles written by the faculty of the Graduate
School of Industrial Administration and published during the 1957-58 academic
year. Single copies may be secured free of charge from: Reprint Editor,
G.S.IJL, Carnegie-Mellon University, Pittsburgh, Pa. 15213. Additional copies
are 50 cents each, unless otherwise noted.
(Continued)
315. Coefficient Estimation in Quadratic Programming Models, by Charles H. Kriebel. Management Science, April 1967.
316. Self-Perception: The Dependent Variable of Human Performance, by Daryl J. Bern,
Organizational Behavior and Human Performance, May 1967.
317. Team Decision Models of an Inventory Supply Organization, by Charles H. Kriebel.
Naval Research Logistics Quarterly, June 1965.
318. Sales Anticipations, Planned Inventory Investment, and Realizations, by Michael C.
Lovell. National Bureau of Economic Research, 1967.
319. Hours of Work Issues by Myron L. Joseph, from Studies prepared for the National
Commission on Technology, Automation, and Economic Progress, February 1966.
320. Marxian Influences in "Bourgeois" Economics, by Martin Bronfenbrenner. The American
Economic Review, May 1967.
321. Cause and Counterfactual, by Herbert A. Simon and Nicholas Rescher. Philosophy of
Science, December 1966.
322. Information Can be Managed, by Herbert A. Simon. Think, May-June 1967.
323. Linear Programming and Optimal Bank Asset Management Decisions, by Kaiman J.
Cohen and Frederick S. Hammer. The Journal of Finance, May 1967.
324. An Empirical Study of Scheduling Decision Behavior, by P. D. Fox and C. H. Kriebel.
The Journal of Industrial Engineering, June 1967.
325. A Guidepost-Mortem, by Martin Bronfenbrenner. Industrial and Labor Relations Review,
July 1967.
326. Tests of a Capital-Theoretic Model of Technological Change, by Robert E. Lucas, Jr.
Review of Economic Studies, 1967.
327. Decision CPM: A Method for Simultaneous Planning, Scheduling, and Control of
Projects, by W. Crowston and G. L. Thompson. Operations Research, May-June 1967.
328. Process Models and Stochastic Theories of Simple Concept Formation, by L. W. Gregg
and H. A. Simon. Journal of Mathematical Psychology, June 1967.
329. Dismissal Pay and Flexible Wage Adjustments: A Theoretical Analysis, by Thomas G.
Moore and Leonard A. Rapping. The Southern Economic Journal, July 1967.
330. Probability Models for Multiprogramming Computer Systems, by D. P. Gaver, Jr.
Journal of the Association for Computing Machinery, July 1967.
331. On the Distinction between Public and Private Goods, by Otto A. Davis and Andrew B.
Whinston. The American Economic Review, May 1967.
332. Economies of Scale in Cash Balances Reconsidered, by Karl Brunner and Allan H.
Meitzer. The Quarterly Journal of Economics, August 1967.
333. Finding a Minimaximal Path in a Disjunctive Pert Network, by E. Balas Theorie Des
Graphes. Journées internationales d'étude, juillet 1966.
334. Piecemeal Policy in the Theory of Second Best, by O. A. Davis and A. B. Whinston.
The Review of Economic Studies, July 1967.
335. Rejoinder to Chase and Hendershott, by Karl Brunner and Allan H. Meitzer. Monetary
Process and Policy, 1967.
336. The Meaning of Monetary Indicators, by Karl Brunner and Allan H. Meitzer. Monetary
Process and Policy, 1967.
337. On the Marxian Capital-Consumption Ratio, by Martin Bronfenbrenner and Yutaka
Kosai. Journal of Science and Society, Fall 1967.
338. Adjustment Costs and the Theory of Supply, by Robert E. Lucas. The Journal of Political Economy, August 1967.
339. Making Effective Use of Computers in Managerial Decision Making, by H. Igor Ansoff.
Automation, October 1967.